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<h1 class="ltx_title ltx_title_section">Difference-of-Gaussians filter</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">The difference-of-Gaussians filter is a feature enhancement algorithm
which acts as a band-pass filter. The <a href="Convolution.html" title="" class="ltx_ref">convolution kernel</a>
is formed as the difference of two Gaussian kernels
</p>
<table id="Sx1.Ex1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.Ex1.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi50.png" width="271" height="21" alt="K_{\mathrm{DoG}}\left(x,y\mid\sigma_{1},\sigma_{2}\right)=K_{\mathrm{G1}}-K_{%
\mathrm{G2}}\,,"></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">where the standard deviation <img id="Sx1.p1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi57.png" width="63" height="16" alt="\sigma_{2}&gt;\sigma_{1}">, and the kernels
are given by <img id="Sx1.p1.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi52.png" width="164" height="21" alt="K_{\mathrm{G1}}=K_{\mathrm{G}}\left(x,y\mid\sigma_{1}\right)">
and <img id="Sx1.p1.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi54.png" width="164" height="21" alt="K_{\mathrm{G2}}=K_{\mathrm{G}}\left(x,y\mid\sigma_{2}\right)">.
The sizes for both kernels are computed as <img id="Sx1.p1.m4" class="ltx_Math" style="vertical-align:-6px" src="mi/mi59.png" width="118" height="21" alt="l=1+2\left\lceil 3\sigma_{2}\right\rceil">.
The standard deviations <img id="Sx1.p1.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi56.png" width="22" height="14" alt="\sigma_{1}"> and <img id="Sx1.p1.m6" class="ltx_Math" style="vertical-align:-5px" src="mi/mi58.png" width="22" height="14" alt="\sigma_{2}"> are user-specified
parameters.</p>
</div>
<div id="Sx1.p2" class="ltx_para">
<p class="ltx_p">Although the kernel <img id="Sx1.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi51.png" width="48" height="18" alt="K_{\mathrm{DoG}}"> is not separable, the filtered
image can be obtained by subtracting two images filtered with <a href="GaussianFilterUI.html" title="" class="ltx_ref">Gaussian kernels</a>
having different standard deviations,</p>
<table id="Sx1.Ex2" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.Ex2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi49.png" width="283" height="18" alt="F=I*K_{\mathrm{DoG}}=I*K_{\mathrm{G}1}-I*K_{\mathrm{G}2}\,."></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Threshold for approximate localization of molecules</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<p class="ltx_p">The <a href="../../detectors/Threshold.html" title="" class="ltx_ref">threshold</a> value can be
specified by users as an expression combining mathematical functions
and operators with variables based on the current raw or filtered
image. Variables provided by this filter are:
</p>
</div>
<div id="Sx1.SSx1.p2" class="ltx_para">
<table class="ltx_tabular ltx_align_middle">
<thead class="ltx_thead">
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<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">DoG.I</span></th>
<th class="ltx_td ltx_align_left">current raw input image</th>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">DoG.F</span></th>
<th class="ltx_td ltx_align_left">corresponding filtered image</th>
</tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">DoG.G1</span></th>
<td class="ltx_td ltx_align_left">result of filtering only with <img id="Sx1.SSx1.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi53.png" width="38" height="18" alt="K_{\mathrm{G1}}">
</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">DoG.G2</span></th>
<td class="ltx_td ltx_align_left">result of filtering only with <img id="Sx1.SSx1.p2.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi55.png" width="38" height="18" alt="K_{\mathrm{G2}}">
</td>
</tr>
</tbody>
</table>
</div>
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<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Filters.html" title="" class="ltx_ref">Image filtering and feature enhancement</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="../../detectors/Detectors.html" title="" class="ltx_ref">Finding approximate positions of molecules</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
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